Dimensions of momentum?

Actually, you are correct in that energy and momentum have different dimensions. What is actually happening is that in the book you are reading, the author is using units (called "natural units") in which the speed of light $c=1$.

The momentum four vector can be written explicitly as $\hat p = [\frac{E}{c},p_i]$.

Click here for more about natural units.

It is common in QFT (and relativity in general) to use natural units in which $c=1$. In SI units, the 4-momentum takes the form $\mathbf p = (E/c, \vec p)$ where $\vec p = \gamma m \vec v$ is the 3-momentum.